Lets say we are given a four components object. To be explicit lets consider that these components are $ x^\mu = \mu $ with $\mu\in{0,1,2,3}$, i.e. $$ x^\mu \sim \left[ \begin{matrix} 0\\ 1\\ 2\\ 3 \end{matrix} \right] .$$ How do we know if these components represent a 4-vector or a spinor? (Forget about the 4-vector typical notation I am using. It's just the notation I choose.) I always read in books that 4-vectors (or tensors in general) are recognized by the way their components transform. Does this also apply to spinors?
Let me expand my question with an "example". Lets consider also some Lorentz transformation parametrized by $\xi_i$ for the "angle" of boosts and $\theta_i$ for the angles of rotations. Suppose we are given the components of this object after the transformation and that they are some array of numbers $y^\mu$. But we are not told how were they calculated or, even more interesting, both $x^\mu$ and $y^\mu$ may were measured. We thus want to relate $x^\mu$ and $y^\mu$.
Suppose that after some 'trial and error' we find that they relate by the linear transformation $$ y^\mu = \left( \exp\left(\frac{i}{2} \omega_{\rho\sigma} \Sigma^{\rho\sigma}\right) \right)^\mu_{\ \ \ \nu} x^\nu $$ (which is nothing more than $y^\mu = \Lambda^\mu_{\ \ \nu} x^\nu$) where $$ \omega_{\rho\sigma} = \left[ \begin{matrix}0 & \xi_{1} & \xi_{2} & \xi_{3}\\ -\xi_{1} & 0 & \theta_{3} & \theta_{2}\\ -\xi_{2} & -\theta_{3} & 0 & \theta_{1}\\ -\xi_{3} & -\theta_{2} & -\theta_{1} & 0 \end{matrix} \right] $$
Is it correct to conclude what follows?
If $\Sigma^{\rho\sigma}$ are given by
then we conclude that the components represent a 4-vector.
If $\Sigma^{\rho\sigma} = \frac{i}{4} [\gamma^\rho,\gamma^\sigma]$ with $\gamma^\mu$ the Dirac matrices, or more explicitly
then we conclude that the components represent a spinor.
If the $\Sigma$'s are different from these but satisfy the Lorentz algebra then the components $x^\mu$ represent other type of object different from a 4-vector or a spinor.
Is this correct? If yes, may this be taken as the definition of a spinor (as happens with 4-vectors) independently for they to satisfy or not Dirac equation?